Just expanded on what the original poster had done, wanted to see it over a longer time period.
Since we were taken over (start of the 08/09 season):
League Cup:
City - 32 games, 15 home, 15 away, 2 neutral. Including 3 Semi-Final ties where you play both home and away.
United - 36 games, 19 homes, 14 away, 2 neutral. Including 4 Semi-Final ties.
Chelsea - 30 games, 14 home, 15 away, 1 neutral. Including 2 Semi-Final ties.
Arsenal - 28 games, 16 home, 11 away, 1 neutral. Including 1 Semi-Final tie.
Liverpool - 34 games, 14 home, 18 away, 2 neutral. Including 4 Semi-Final ties.
Spurs - 28 games, 14 home, 12 away, 2 neutral, Including 2 Semi-Final ties.
FA Cup (replays are separate from the home and away stats):
City - 35 games, 14 home, 12 away, 3 home replays, 1 away replay, 5 neutral.
United - 36 games, 14 home, 14 away, 2 home replays, 2 away replays, 4 neutral.
Chelsea - 45 games, 17 home, 14 away, 3 home replays, 2 away replays, 9 neutral.
Arsenal - 44 games, 20 home, 12 away, 2 home replays, 3 away replays, 7 neutral.
Liverpool - 30 games, 11 home, 10 away, 2 home replays, 4 away replays, 3 neutral.
Spurs - 31 games, 14 home, 10 away, 3 home replays, 1 away replay, 3 neutral.
So since the takeover we haven't been that hard done by, I think Liverpool have had it worse in the domestic cups, whilst Arsenal have clearly benefitted.
Worth pointing out in though that from the start of the 2008 season to the end of the 2014 season we were 14 home draw and 5 away in the FA Cup, which means we're 0 home and 7 away in the last 3 seasons.
So early on we benefitted a little bit but in the last 3 seasons it's been pretty brutal.